共形双曲度量的孤立奇点

Nitsche证明了共形双曲度量的孤立奇点要么是锥奇点,要么是尖奇点,二者必其一(Nitsche J.über die isolierten singularit?ten der L?sungen von△u=e~u J]. Math Z, 1957, 68(3):316-324.).本文利用展开映射证明了在孤立奇点附近存在复坐标z,使得度量要么为(4α~2|z|~(2α-2))/((1-|z|~(2α))~2)|dz|~2,其中α 0,要么为|z|~(-2)(ln|z|)~(-2)|dz|~2.

Isolated Singularities of Conformal Hyperbolic Metrics

Nitsche proved that an isolated singularity of a conformal hyperbolic metric is either a conical singularity or a cusp one (Nitsche, J., {\"{U}}ber die isolierten singularit{\"{a}}ten der L{\"{o}}sungen von $\Delta u=\rme^{u}$, {\it Math. Z}, 1957, vol.68, no.3, pp.316--324.). The authors prove that there exists a complex coordinate $z$ centered at the singularity where the metric has the expression of either $\displaystyle{\frac{4\alpha^2\vert z \vert^{2\alpha-2}} {(1-\vert z \vert ^{2\alpha})^2}\vert \mathrm{d} z \vert^2}$ with $\alpha>0$ or $\displaystyle{\vert z \vert ^{-2}\big(\ln|z|\big)^{-2}|\rmd z|^{2}}$ by developing map. \\